Chapter 4
The Nash Demand Game Revisited: a
Logit-Equilibrium Approach
First version: August 2000
This version: March, 2001
Abstract
We prove existence of a logit equilibrium in the Nash Demand Game. If
both bargainers are sufficiently ‘irrational,’ (i.e., if their error parameters
are large enough), this equilibrium is unique. If, moreover, they have the
same utility function, it is also symmetric. We then compute examples
of logit equilibria in the Nash Demand game for various speciÞcations of
bargainers’ utilities and error parameters. We focus on error parameters
that were shown to be empirically relevant in a different context. Although
we consider error parameters for which we were not able to show uniqueness of a logit equilibrium theoretically, in no example do we Þnd multiple
logit equilibria. Our computations accord well with the intuition that in
a boundedly rational world it does not pay off to bargain aggressively:
the danger of losing the ‘cake’ is greater than the loss from leaving a part
of the cake unclaimed. Our computations strongly support the hypothesis that as bargainers’ error parameters tend to zero at the same rate,
the logit-equilibrium density functions converge to the Nash bargaining
solution.
JEL ClassiÞcation: C70, C78, D81
Keywords: Nash Demand Game, bargaining, logit equilibrium, probability
choice rule, Nash bargaining solution
88
1
Introduction
This note is concerned with a boundedly rational analysis of the Nash Demand
Game (NDG). The NDG was proposed by Nash (1953) as a non-cooperative
model of bargaining between two players. It describes a situation in which both
players announce simultaneously their take-it-or-leave-it demands. If these demands do not exceed the overall size of the available ‘cake,’ each player receives
his claim; otherwise, both players receive nothing.
The NDG possesses a continuum of equilibria, all of them Pareto-efficient.
Bargaining theory has focused primarily on the selection among these. As
Binmore et al. (1998) pointed out in a different context, however, the selection
among multiple Pareto-efficient bargaining outcomes presupposes the solution
of the more fundamental problem of how players coordinate on a Pareto-efficient
outcome in the Þrst place. Each Pareto-efficient outcome, after all, involves
precarious balancing on the brink: if either party were to demand but a fraction
of the cake more, an agreement would not be reached. In other words, each
equilibrium in the NDG has the property that the loss incurred by deviating
from it is highly asymmetric with respect to the direction of the deviation:
demanding unilaterally a bit more than one’s equilibrium share is far more
costly than demanding a bit less.
While the precarious nature of Nash equilibria in the NDG does not matter
in the world of perfectly rational agents, it might matter in a world that is less
accommodating. To examine this intuition, we allow the bargaining players to
make mistakes. SpeciÞcally, we suppose that each player, rather than selecting
his demand with certainty, follows a “probabilistic choice rule” (Luce, 1959),
with the probability of each demanded share depending on the expected utility
from this share. There are at least three interpretations of such probabilistic
choice behavior, which we review in section 2.1.
In a strategic context, such as the NDG, a player’s expected utility from a
claim depends on his belief about the other player’s claim(s). If each player’s
belief matches the actual choice probabilities of his opponent (determined, in
turn, by the opponent’s beliefs about the Þrst player), the resulting choice
probabilities reinforce each other. If this is true, the situation constitutes a
“boundedly rational” equilibrium. This equilibrium was Þrst introduced by
McKelvey and Palfrey (1995), who coined it a quantal response equilibrium
and established its existence in Þnite-strategy normal-form games.
A special case of a quantal response equilibrium in which the probability
of choosing each strategy is an exponential function of the expected utility associated with this strategy is called a logit equilibrium. It arises under the
89
assumption that noise (players’ mistakes in calculations, etc.) is distributed
according to the double exponential function. Besides having the intuitive theoretical property that players’ choice probabilities are unaffected by adding a
constant to their payoffs, logit equilibrium is convenient to work with because
various degrees of ‘irrationality’ (proclivity to mistakes or randomizing) may
be modeled through varying a single parameter (the so-called error parameter).
McKelvey and Palfrey (1995) proved that as players’ error parameters tend to
zero (i.e., as players become more and more perfectly rational), logit equilibrium converges to Nash equilibrium. Logit equilibrium thus may be thought of
as a boundedly rational generalization of Nash equilibrium.
Logit equilibrium has been championed by Goeree and Holt (see, e.g., Goeree and Holt, 1999; Anderson, Goeree and Holt, 1998, 2001, forthcoming 1,
fortcoming 2; Capra, et al., 1999) as a tool for explaining seemingly anomalous
behavior of experimental subjects in economic games. Goeree and Holt found
a strong correspondence between logit equilibrium predictions for minimum effort coordination games, all-pay auctions, voluntary contribution games, etc.
and the laboratory data available on these games. Although we feel there are
some problems with interpreting their Þndings (see section 2.4), we believe they
collected enough evidence to justify further research on logit equilibrium (and
quantal response equilibrium in general), whether on properties of this equilibrium, circumstances under which it may arise or applications of this equilibrium
to particular games.
This note is an extension of the theoretical research of Goeree and Holt
to the NDG. After we introduce and discuss the concepts of quantal response
equilibrium and logit equilibrium (see section 2), we deÞne logit equilibrium in
the NDG and prove its existence. If players are sufficiently ‘irrational’ (i.e., if
their error parameters are large), we prove the uniqueness of this equilibrium.
If, moreover, both players have the same utility, we prove that (the unique) logit
equilibrium must be symmetric. Finally, we compute a few examples of logit
equilibria for various speciÞcations of players’ utilities and error parameters and
discuss their properties. In no case were we able to Þnd more than one logit
equilibrium, although all considered error parameters lay outside the range of
theoretically established uniqueness. This supports our belief (based on results
of Goeree and Holt for games similar to the NDG) that there is indeed a unique
logit equilibrium in the NDG regardless of players’ error parameters, although
we were not able to prove so in general. All examples reßect the asymmetric
nature of Nash equilibria in the NDG mentioned above: players indeed ‘fear’
to bargain aggressively and so put more weight on small claims.
Finally, the examples point decisively toward the conclusion that as players’
90
error parameters go to zero at the same rate (i.e., as players become more and
more perfectly rational at the same rate), logit equilibrium density functions
converge to the Nash bargaining solution. This Þnding corroborates the relevance of the Nash bargaining solution when noise is added to the basic setup of
the NDG (cf. also Young, 1993; and Binmore and Dasgupta, 1987, chapter 4).
2
2.1
Logit Equilibrium: Reviewing the Concept
Probabilistic Choice
Consider an agent who has to make a choice between two actions A1 and A2 . He
believes that A1 has expected utility U e (A1 ) and A2 expected utility U e (A2 ). A
perfectly rational agent, who is absolutely sure about the accuracy of his assessment of U e (A1 ) and U e (A2 ), would opt for an action with the higher expected
utility. Agents, however, may not always be perfectly rational. To model the
departure from perfect rationality, suppose that a small degree of noise affects
the agent’s comparison of expected utilities. Then, the agent chooses the action
with lower expected utility with non-zero probability. Algebraically,
P (A1 ) = Pr (U e (A1 ) − U e (A2 ) > µε) ,
(1)
where P (A1 ) is the probability of choosing action A1 , ε is a random variable
with zero mean, and µ is a positive real number. The distribution of ε describes
the qualitative nature of the noise, while µ captures–in a sense to be made
precise later–its signiÞcance. We call µ the agent’s error parameter.
There are at least three interpretations of equation (1). First, the agent’s
preferences are subject to small random shocks. The agent always behaves in a
perfectly rational manner but, his preferences being volatile and unobservable,
he appears to the external observer as making mistakes in judgement. An
example of such volatile preferences might be varying risk aversion. If action
A1 leads to a higher, but less certain, expected payoff than action A2 , then the
agent may choose A1 with high probability when feeling elated and A2 when
feeling dejected.
Second, the agent’s preferences are stable, but he is unable to calculate
e
U (A1 ) and U e (A2 ) precisely. An auctioneer who has to make calculations
under severe time constraints or who sometimes mishears the price quoted may
serve as an example here. This interpretation of (1) is perhaps the easiest
to imagine. In what follows, we refer to this interpretation as mistakes in
computation.
Third, the agent has no conscious utility function, but he is, nonetheless,
91
inßuenced by preferences buried in his subconscious mind. This interpretation
was suggested by Chen, Friedman and Thisse (1997).
As µ becomes smaller, the agent’s decision making becomes less noisy and
equation (1) approaches the perfectly rational ideal. As µ grows larger, the
error term µε starts to dominate the difference between U e (A1 ) and U e (A2 ).
Consequently, both actions A1 and A2 will be selected with almost equal probabilities. Hence, by varying µ one is able to capture different degrees of imperfect
rationality.
Equation (1) formalizes psychologist Luce’s (1959) notion of “probabilistic
choice.” Depending on the speciÞcation of ε, this equation generates a number of different models of boundedly rational behavior. If, for instance, ε is
distributed uniformly on [−1, 1], then (1) turns into
P (A1 ) =





1
2
+
U e (A1 )−U e (A2 )
2µ
if |U e (A1 ) − U e (A2 )| < µ;
if U e (A1 ) − U e (A2 ) ≤ −µ;
if U e (A1 ) − U e (A2 ) ≥ µ.
0
1
(2)
Here the agent randomizes between both actions, his choice probabilities being proportional to the difference in corresponding expected utilities unless this
difference is too large. In the latter case, he chooses with certainty the action
yielding the higher expected utility. Equation (2 ) is known as a linear probability choice model, applied Þrst in a game-theoretical context by Rosenthal
(1989).
If ε is drawn from a double-exponential distribution,1 then (1) generates
choice probabilities of a “logit” form
P (Ai ) =
exp
2.2
³
exp
U e (A1 )
µ
³
´
U e (A1 )
µ
+ exp
´
³
U e (A2 )
µ
´ , i = 1, 2.
(3)
Quantal Response Equilibrium and Logit Equilibrium in
Finite-Strategy Games
Let us now switch from the simple decision problem of a single agent to a strategic situation involving n agents, n ≥ 2. From now on we shall call these agents
players and their actions strategies. While we still assume that each player
i = 1, . . . , n behaves according to the probabilistic choice rule (1), properly
generalized for more strategies at hand, the expected utilities that determine
1
Double-exponential distribution (also called extreme value distribution) is deÞned by the
cumulative density function F (ε) = exp(− exp(ε)).
92
the particular form of this rule now depend on player i’s beliefs about other
players’ strategies.
To see this, let us adopt the mistakes-in-computation interpretation. Each
player holds beliefs about his opponents’ strategies and attempts to choose the
best response to these beliefs: he calculates the expected utility from each of
his pure strategies and adopts the strategy with the highest utility. Since his
calculations are noisy, however, this strategy is unlikely to be the actual best
response to his beliefs. Think of this pure strategy as a realization of a mixed
strategy–a ‘noisy best response.’
Assume that every player is hard-wired to play a particular noisy best response. If a player’s noisy best response differs from what other players believe
about him, these players are likely to revise their beliefs. If, in contrast, each
player’s noisy best response matches exactly what other players believe about
him, an equilibrium emerges. This equilibrium was introduced by McKelvey
and Palfrey (1995), who coined it a quantal response equilibrium and proved its
existence in n-player normal-form games with a Þnite number of strategies.
Let us review and slightly reÞne the formal deÞnition of this equilibrium.
Consider a Þnite game Γ with n players; N = {1, . . . , n}. Each player i ∈ N
Q
has a strategy set Ai = {ai1 , . . . , aiJi ); A ≡ i Ai . Each player i ∈ N has the
(von Neumann-Morgenstern) utility function Ui : A → R. For each i ∈ N, ∆i
Q
denotes the set of probability measures on Ai , and ∆ ≡ i ∆i .2 Finally, for
each i ∈ N, βi ∈ ∆−i describes player i’s beliefs about other players’ strategies.
Player i’s expected utility from the mixed strategy pi ∈ ∆i , pi = (pi1 , . . . , piJi ),
subject to his beliefs βi is
Uie (pi | βi ) =
X
j
pij
X
a−i ∈A−i
βi (a−i )Ui (aij , a−i ).
Player i’s best response ρ∗i (βi ) to βi is deÞned as
ρ∗ (βi ) ≡ {p ∈ ∆i | Uie (p | βi ) ≥ Uie (e
p | βi ) for all pe ∈ ∆i }.
NASH EQUILIBRIUM: A vector p = (p1 , . . . , pn ) ∈ ∆ is a Nash equilibrium if
1. For all i ∈ N, βi = p−i and
2. For all i ∈ N, pi ∈ ρ∗i (βi ).
2
In what follows, we identify the pure strategy aij with the mixed strategy that places
probability one on aij , and hence understand aij as an element of ∆i .
93
Suppose that each player i ∈ N, when deciding what pure strategy to use,
attempts to calculate Uie (aij | βi ) for all aij ∈ Ai . Unfortunately, he makes a
computational mistake and instead of Uie (aij | βi ) obtains
b e (aij | βi ) = U e (aij | βi ) + εij ,
U
i
i
the second term capturing the mistake. Let εi ≡ (εi1 , . . . , εiJi ) be distributed
according to the joint density function fi , such that for each j ∈ Ji the marginal
density function exists, and E(εij ) = 0. Suppose that player i ∈ N plays the
b e (aij | βi ), i.e., he
best response as determined by his ‘noisy’ calculations of U
i
plays strategy aij if and only if
b e (aij | βi ) + εij ≥ U
b e (aik | βi ) + εik ,
U
i
i
(4)
for all k ∈ Ji . DeÞne Rij (βi ) as a set of proÞles εi for which the above relationship holds. Then, ρi (βi ) = (ρi1 (βi ), . . . , ρiJi (βi )) ∈ ∆i deÞned as
Z
ρij (βi ) =
f(ε)dε
Rij (βi )
determines player i’s noisy best response to his beliefs βi , as described verbally
above. Quantal response equilibrium is a proÞle of noisy best responses that
are consistent with beliefs that induce them. Formally,
QUANTAL RESPONSE EQUILIBRIUM: A vector p = (p1 , . . . , pn ) ∈
∆ is a quantal response equilibrium (QRE) if
1. For all i ∈ N, βi = p−i , and
2. For all i ∈ N, pi = ρi (βi ).
Comparison of this deÞnition with that of Nash equilibrium reveals a strong
link between both concepts.
If for some i ∈ N, the random variables εij , j ∈ Ji , are independent and
drawn from the same double exponential distribution with cumulative density
function
³
³ ´´
F (ε) = exp − exp µεi ,
(5)
player i’s noisy best response becomes the logit best response resembling equation (3):
³ e
´
U (a | β )
exp i µiji i
³ e
´.
ρij (βi ) = P
Ui (aij | βi )
Ji
exp
k=1
µi
94
We call µi player i’s error parameter.
Logit equilibrium is a QRE in which all players play the logit best responses.
Formally,
LOGIT EQUILIBRIUM: A vector p = (p1 , . . . , pn ) ∈ ∆ is a logit equilibrium if
exp
ρij (p−i ) = P
Ji
³
U e (aij | p−i )
µi
k=1 exp
³
´
U e (aik | p−i )
µi
´ , for all i ∈ N, j ∈ Ji .
(6)
McKelvey and Palfrey (1995) proved that if all µi → 0 at the same rate, the
logit equilibrium converges to a Nash equilibrium of the game Γ. If, however, all
µi → +∞, players’ logit equilibrium responses degenerate to uniform randomization. Logit equilibrium therefore may be thought of as a boundedly rational
generalization of Nash equilibrium, with the size of players’ error parameters
determining the distance from the latter.
A drawback of QRE is that it is, just as Nash equilibrium, a static concept.
Moreover, QRE is an equilibrium in probability measures over players’ strategy
sets and hence it is more complex than a pure strategy Nash equilibrium (about
as complex as a mixed-strategy Nash equilibrium). An important question is,
therefore, how players are going to coordinate on a QRE. This question was
addressed by Chen, Friedman and Thisse (1997), who investigated a boundedly
rational learning process similar to Þctitious play. In this process, players are
matched in successive periods, with each player playing a particular type of
noisy best response to the empirical distribution of his opponents’ strategies
derived from past plays. The authors show that if players are not too close to
‘perfectly rational men’ (i.e., in our notation, if players’ error parameters are not
too small), this learning process converges to a QRE.3 They do not, however,
characterize quantitatively the size of players’ error parameters required for
the convergence, except in an example (see Chen, Friedman and Thisse, 1997,
p. 49). Since their proof of convergence is based on a the same ‘contraction’
argument as our proof of uniqueness of logit equilibrium in the NDG (see section
3) and since our proof works only for relatively large error parameters, our
conjecture is that the learning process proposed by these authors also converges
only for relatively large error parameters. Dynamic justiÞcation of QRE in a
Þnite-strategy game for plausibly small error parameters still needs to be found.
3
One cannot expect the process to converge for all games if players are almost perfectly
rational (i.e., if their noisy best responses are almost best responses), because then the process
becomes very close to traditional Þctitious play, which does not converge in general (see
Shapley, 1964).
95
2.3
Logit Equilibrium in Games with a Continuum of Strategies
Lopez (1995) extended logit equilibrium to games with a continuum of strategies. Consider a symmetric n-player game Γ with a continuous strategy set
[a, b] ⊂ R common to all players. A proÞle F = (F1 , . . . , Fn ) of cumulative density functions Fi deÞned on [a, b] constitutes a logit equilibrium in this game, if
it solves a system of equations that are continuous analogues of (6):
fi (x) = Fi0 (x) = Z
a
exp
b
³
exp
Uie (x | F−i )
µi
³
´
Uie (s | F−i )
µi
´
ds
, for all i ∈ N, x ∈ [a, b].
(7)
Here fi is a density function corresponding to Fi , F−i a proÞle of Fj ’s for all
j 6= i, and Uie (x | F−i ) player i’s utility from playing x, given that his beliefs
about other players coincide with F−i .4 As before, µi stands for player i’s error
parameter. Note that for all i, the denominator in (7) is a constant guaranteeing
that fi integrates to one on [a, b].
In practice, logit equilibrium (7) is typically found by solving a system of
differential equations obtained by differentiating (7) with respect to x. As can
be easily veriÞed, this system is given as
µi Fi00 (x) = Fi0 (x)Uie0 (x | F−i ) s.t. Fi (a) = 0, Fi (b) = 1
(8)
for all i ∈ N, x ∈ [a, b].5 It is important to note, however, that systems (8) and
(7) are not equivalent. Since system (7) may have solutions that are not twice
differentiable, some logit equilibria may not be found by solving system (8).
However, if system (7) can be shown to have only twice differentiable solutions,
then any logit equilibrium F is necessarily a solution to (8) as well.
Note that if µi → 0, the i-th equation in system (8) approaches
fi (x)Uie0 (x | F−i ) = 0 s.t. Fi (a) = 0, Fi (b) = 1.
This means that the logit-equilibrium density function fi converges to the degenerate density function fi∞ such that fi∞ (x) = 0 whenever Uie0 (x | F−i ) 6= 0.6
4
Equation (7) does not give a rigorous deÞnition of logit equilibrium for games with a
continuum of strategies, but rather a characterization of such an equilibrium. For a rigorous
deÞnition, one would Þrst need to show how each player i’s logit response (fi ) can be derived
from his mistakes in expected utility computations, just as we did in the previous section.
We are aware of no such deÞnition. This makes us wonder whether the continuum of random
variables involved in such a derivation is not a more fundamental problem than it Þrst seems.
5
Here prime denotes the Þrst and a double prime denotes the second derivative with respect
to x.
6
Note that if Ui0 (x) = 0 on a set of measure zero only, then fi∞ is a linear combination of
96
In other words, fi∞ can only place positive probability on strategy x for which
Uie0 (x | F−i ) = 0. If for any such x, Uie00 (x | F−i ) < 0, then fi∞ is the player
i’s best response to F−i . Suppose that this is true for all i ∈ N. Then, as all
µi → 0, logit equilibrium must converge to an interior Nash equilibrium of the
game Γ, given that such equilibrium exists.
If µi → +∞, the i-th equation in (8) approaches
Fi00 (x) = 0 s.t. Fi (a) = 0, Fi (b) = 1.
In this case, the logit-equilibrium density function fi converges to the uniform
density function.
If all µi ’s are of intermediate size, the logit-equilibrium density functions
are between the fi∞ (which correspond to an internal Nash equilibrium) and
uniform density functions. Hence, even in the continuous case, logit equilibrium
may be understood as a boundedly rational generalization of Nash equilibrium.
Anderson, Goeree and Holt (1997) proposed a boundedly rational learning
process that converges to a logit equilibrium in a wide class of games that
feature a continuum of strategies (the so-called potential games). The process
arises if the game is played repeatedly, and if in each period players attempt to
adjust their strategy optimally in the direction of increasing payoff, but make
normally distributed mistakes. Convergence is independent of players’ error
parameters. What diminishes the beauty of this result are the strong epistemic
foundations of the process: in each period each player knows precisely the mixed
strategies of all other players, although he observes mere realizations of these.
Another important question is whether this process represents the limit of some
plausible learning process in a Þnite-strategy game. In particular, a connection
to the learning process studied by Chen, Friedman and Thisse (1997) (see the
previous section) seems worth examining.
2.4
Logit Equilibrium: A Perspective on Intuitive Behavioral
Anomalies?
As mentioned in the introduction, logit equilibrium has been recently defended
by Goeree and Holt (e.g., Goeree and Holt, 1999 or Anderson, Goeree and Holt,
forthcoming 1 and forthcoming 2) as a means of reconciling game theory with
‘anomalous’ experimental results (i.e., results that go contrary to the Nash equilibrium predictions). The authors calculated logit equilibrium in more than ten
symmetric games with a continuum of strategies (minimum effort coordination
Dirac delta functions δx0 (·), well-known from theoretical physics. Dirac delta function δx0 (·)
is a generalized function that is zero everywhere except for x0 , inÞnite at x0 , and integrates
to one on [a, b].
97
games, all-pay auctions, voluntary contribution games, etc.) and compared
this equilibrium with available experimental data. Their Þndings were quite
compelling. In all games the average logit-equilibrium strategy was almost the
same as the average strategy played in the few last rounds of the experiment.
In particular, both the average logit-equilibrium strategy and the average actually played strategy were highly sensitive to a certain underlying parameter of
the game (e.g. cost of effort in the minimum effort coordination game), whilst
the Nash equilibrium strategy displayed no such sensitivity. In the Traveler’s
Dilemma game the logit equilibrium density function traced quite accurately
the experimentally observed distribution of strategies (see Anderson, Goerre
and Holt, forthcoming 1).
Although Goeree and Holt interpreted these results with great zeal, we feel
some caution is necessary. One problem is the aggregate nature of the data.
Logit equilibrium, as explained above, arises when each player in a game plays
the logit best response. This, however, was not what Goeree and Holt observed
in their experiments. Even in the Traveler’s Dilemma, the most convincing
of their experiments, they observed merely that the distribution of strategies
among experimental subjects had a form of the logit best response. This distribution of strategies can arise even if no experimental subject actually playes
the logit best response. To justify the comparison between logit best response
and the distribution of strategies among players, one needs an interpreation
of logit equilibrium as a population steady state, an interpretation similar to
that of Nash equilibrium as an evolutionarily stable set (Maynard Smith and
Price, 1973). Without this interpretation, it is difficult to conclude what the
celebrated agreement between theory and the data means. Providing such interpretation, however, is not the aim of the present paper.
Even if Goeree and Holt’s observations are a consequence of each experimental subject playing the logit best response, not all difficulties with interpreting
their Þndings are resolved. The main problem is that no interpretation of the
probabilistic choice rule (and hence of the logit best response) is very plausible
in such a simple strategic situation as a laboratory experiment. Since Goeree and Holt’s experimental subjects were motivated by monetary gains, their
preferences over outcomes were by deÞnition very simple. And, since all experiments took only a relatively short time, the subjects’ attitude to risk could not
vary much. The “random preference” interpretation of the probabilistic choice
rule is thus of little avail, as is the “subconscious utility” interpretation. Only
the “mistakes in computation” interpretation seems viable, but to make this
interpretation sound in a strategic context, one would need to relate it to the
formation of subjects’ beliefs about each other.
98
In spite of the difficulties with interpreting Goeree and Holt’s Þndings (and
with justifying logit equilibrium by means of a learning process), we think
that logit equilibrium and the like concepts represent a promising direction for
the game theory. The application of logit equilibrium to the NDG, a game
so far analyzed almost exclusively by traditional game-theoretic tools, is our
contribution to the research in this vein.
3
Logit Equilibrium in the Nash Demand Game
We now establish existence of logit equilibrium in the NDG (Nash, 1953), and
prove its uniqueness and symmetry for large error parameters.
Recall that in the NDG two players bargain over a cake, announcing their
demands simultaneously. If the sum of the demands does not exceed the total
size of the cake, each player gets what he wants; otherwise, both players get
nothing. The game has a continuum of Nash equilibria in pure strategies,
namely all outcomes where players’ demands sum up to the size of the cake.
Suppose that the cake has unit size. Suppose also that player i = 1, 2,
has an error parameter µi , and a utility function Ui that is non-decreasing,
bounded and non-negative on [0, 1].7 Denote by Fi , i = 1, 2, the cumulative
density function that governs player i’s demand and by fi the corresponding
density function. Then, the expected utility player i derives from demanding x
is
Uie (x) = Ui (x) Pr (player j 6= i demands no more than 1 − x) =
= Ui (x)F3−i (1 − x).
If a pair of cumulative density functions F = (F1 , F2 ) constitutes a logit equilibrium in the NDG, it must satisfy
fi (x) = Fi0 (x) = Z
0
exp
1
³
exp
Ui (x)F3−i (1−x)
µi
³
´
Ui (s)F3−i (1−s)
µi
´
, for i = 1, 2.
(9)
ds
The following theorems assure that for any reasonable pair of utilities U1
and U2 , logit equilibrium exists and, if both µi are large enough, it is unique.8
7
The assumption that Ui , i = 1, 2, is non-negative on [0, 1] is adopted only because it
simpliÞes the ensuing analysis.
8
Theorem 1 draws heavily on Proposition 1 in Anderson, Goeree and Holt (forthcoming 2).
This proposition establishes the existence of logit equilibrium in all games with a continuum
of strategies [x, x] in which each player’s expected utility (payoff) from playing strategy x
99
THEOREM 1: The NDG has a logit equilibrium F = (F1 , F2 ), both cumulative functions F1 , F2 being differentiable. If U1 and U2 are continuous
on [0, 1], then so are f1 and f2 . If, moreover, U1 and U2 are differentiable
m-times on [0, 1], 1 ≤ m ≤ +∞, then both F1 and F2 are differentiable
m + 1 times (inÞnitely many times if m = +∞ ).
Proof: Let M be deÞned as a set of continuous cumulative density functions
on [0, 1], i.e.
M ≡ {F ∈ C[0, 1] | F non-decreasing, F (0) = 0, F (1) = 1}.
Fix a pair of µ1 , µ2 ∈ R+ and deÞne the integral operator T : M×M → M×M
as
where
T (F1 , F2 ) = (Te1 (F2 ), Te2 (F1 )),
Z
x
Tei (F )(x) = Z0 1
0
exp
exp
³
³
Ui (s)F3−i (1−s)
µi
Ui (s)F3−i (1−s)
µi
for i = 1, 2. Then, system (9) may be rewritten as
(F1 , F2 ) = T (F1 , F2 ),
´
´
(10)
ds
,
(11)
ds
(12)
and establishing the existence of logit equilibrium in the NDG is equivalent
to showing that T has a Þxed point. By Schauder’s second theorem (see, e.g.,
Smart, 1974, p. 25), T has a Þxed point if (i) its range of values H = {T (F1 , F2 )
| (F1 , F2 ) ∈ M× M} is compact, and (ii) T is a continuous mapping.9 We shall
next prove points (i) and (ii).
Point (i): Note Þrst that H may be written as K1 × K2 , where Ki ≡ {Tei (F ) |
F ∈ M} is a range of values of operator Tei . Since a Cartesian product of two
compact sets is a compact set, it suffices to prove that both sets Ki , i = 1, 2,
are compact. We do this just for K1 ; compactness of K2 may be established
analogously.
is a continuous function of the distribution (density, cumulative density) functions of other
players. Since it was not clear to us whether the authors meant the continuity with respect to
the distribution functions of other players evaluated at x, or with respect to the distribution
functions as elements of C[x, x], and since we suspect that in the latter case–which would
cover the NDG–the proposition may generally fail to hold, we preferred to prove the existence
of logit equilibrium in the NDG ourselves.
9
Here compactness and continuity are understood with respect to the norm in C[0, 1],
kF ksup = supx∈[0,1] F (x).
100
To show that K1 is compact, we take advantage of the following proposition
by Arzela (taken from Lusternik and Sobolev, 1961, p. 61):
Proposition (Arzela): A set K ⊂ C[0, 1] is compact if and only if the elements of K are (a) uniformly bounded and (b) equicontinuous.
Uniform boundedness of K1 is evident since K1 ⊂ M and elements of M are
uniformly bounded by deÞnition. Equicontinuity of K1 requires us to show that
for
¯ exists δ(ε) > 0 such that for any F ∈ M it holds that
¯ any given ε > 0, there
¯
¯e
0
¯T1 (F )(x) − Te1 (F )(x )¯ < ε, whenever |x − x0 | < δ(ε), with x, x0 ∈ [a, b]. Since
¯
¯
¯e
¯
¯T1 (F )(x) − Te1 (F )(x0 )¯ =
≤
this condition is met for δ(ε) = exp
³
¯
¯Z 0
¯ x
³
´ ¯
¯
¯
U1 (s)F2 (1−s)
exp
ds¯
¯
µ
1
¯
¯ x
≤
Z 1
³
´
U1 (s)F2 (1−s)
exp
ds
µ1
0
³
´
|x − x0 |
exp U1µ(1)
1
³
´
,
exp U1µ(0)
(1
−
0)
1
U1 (0)−U1 (1)
µ1
´
ε.
Point (ii): T is continuous on M × M if and only if each Tei , i ∈ 1, 2, is
continuous on M. We prove continuity of Te1 ; the continuity of Te2 may be
established analogously.
To prove continuity of Te1 on M, choose an arbitrary function F ∈ M and
kFk − F ksup = 0.
consider any sequence {Fk }+∞
k=0 , Fk ∈ M,°such that limk→+∞
°
°
°
Then, Te1 is continuous at F if limk→+∞ °Te1 (Fk ) − Te1 (F )°
= 0. Given that
sup
F is arbitrary, this result guarantees that Te1 is continuous on M.
From (11) it follows that for any x ∈ [0, 1], 10
¯
¯
¯
¯e
¯T1 (Fk )(x) − Te1 (F )(x)¯ ≤
¯Z x F U Z 1 F U
Z x FU Z 1 F U ¯
¯
k 1
1
1
k 1¯
¯
e µ1
e µ1 −
e µ1
e µ1 ¯¯
¯
0
0
≤
≤
Z0 1 F U Z 10 F U
1
k 1
µ
µ
e 1
e 1
0
0
¯Z x µ F U
¶ Z 1 FU
¶¯
Z x FU Z 1 µ F U
F U1
F U1
¯
¯
k 1
1
1
k 1
µ
µ
µ
µ
µ
µ
¯
¯
1
1
1
1
1
1
−e
−
−e
e
e
e
e
¯
¯
0
0
0
0
≤
. (13)
Z 1 FU Z 1 F U
k 1
1
µ
µ
e 1
e 1
0
10
0
The arguments of integrands were dropped here for the sake of brevity.
101
Noting that for any A, B ∈ R, |A − B| ≤ |A| + |B|, and applying this inequality
to the numerator of (13), we arrive at
where
³
´
U1 (1)
¯
¯
2
exp
µ1
¯e
¯
¯T1 (Fk )(x) − Te1 (F )(x)¯ ≤ h
³
´
i2 Bk ,
exp U1µ(0)
(1
−
0)
1
Bk =
Z
0
(14)
¯
¶
µ
¶¯
µ
¯
¯exp U1 (s)Fk (1 − s) − exp U1 (s)F (1 − s) ¯ ds.
¯
¯
µ1
µ1
1¯
But, since for any U ≥ 0, µ > 0, and Y, Y 0 ∈ [0, 1], it holds that
¯
¶
µ
¶¯
µ ¶
¸
µ ¶·
µ
0
¯
¯ 0
¯
UY ¯¯
U
U
¯Y − Y ¯ ,
¯exp UY
−
exp
≤
exp
exp
−
1
¯
¯
µ
µ
µ
µ
we have
µ
U1 (1)
0 ≤ Bk ≤ exp
µ1
¶·
µ
¶
¸
U1 (1)
exp
− 1 kFk − F ksup .
µ1
(15)
This Þnishes the proof of continuity of Te1 .
The differentiability of (F1 , F2 ) that solve (12) follows from the equivalence
between systems (12) and (9): since for any i = 1, 2, the right-hand side of (9)
is a well-deÞned function of F3−i , the derivative Fi0 on the left-hand side of (9)
is well-deÞned as well. If functions Ui , i = 1, 2, are continuous, then fi = Fi0 ,
i = 1, 2, given by (9), are continuous transformations of continuous functions
F1 and F2 and hence themselves continuous. Finally, if Ui , i = 1, 2, are once
differentiable, then both f1 and f2 are differentiable transformations of F1 and
F2 which are also differentiable. Hence, for i = 1, 2, fi can be differentiated,
and so Fi possesses the second order derivatives. Higher order differentiability
of Fi , i = 1, 2, may be established recursively in like manner.
¥
THEOREM 2: If µ1 and µ2 are large enough, logit equilibrium in the NDG is
unique. Given that U1 (0) = U2 (0) = 0, ‘large enough’ means µi > θUi (1),
where

 r
´
³
√
1
2
1 .
(16)
+ =
θ = ln−1  3 62 + 6 105 + q¡
√
¢
6
3
3 3 62 + 6 105
.
= 3.84.
Proof: From Theorem 1 we know that system (9) has a solution which is a
Þxed point of operator T deÞned by (10) and (11). This Þxed point is unique
102
if operator T is a contraction mapping. By deÞnition of T , this holds if and
only if both operators Te1 and Te2 , deÞned by (11), have this property. As for
Te1 , formulae (14) and (15) imply that for any pair of functions F , G ∈ M, it
holds that
¯ 2 exp
¯
¯
¯e
¯T1 (F )(x) − Te1 (G)(x)¯ ≤
³
i
´h
³
´
−
1
exp U1µ(1)
1
³
´
kF − Gksup , (17)
2U1 (0)
exp
µ1
2U1 (1)
µ1
and hence Te1 is a contraction whenever
2 exp
³
´h
³
´
i
exp U1µ(1)
−
1
1
³
´
< 1.
2U1 (0)
exp
µ1
2U1 (1)
µ1
(18)
Since the limit of the left-hand side of (18) for µ1 → +∞ equals zero, (18) is
satisÞed for any large enough µ1 . If U1 (0) = 0, then the left-hand side of (18) is
and so (18) holds whenever U1µ(1)
< 1θ , where θ is
an increasing function of U1µ(1)
1
¡ 1 ¢1 ¤
¡2¢ £
a root of 2 exp θ exp θ − 1 = 1. Condition (16) rephrases this statement
explicitly.
Performing the analogous exercise for Te2 Þnishes the proof.
¥
COROLLARY: Suppose that both players have the same utility function U,
U(0) = 0, and the same error parameter µ such that µ > θU(1), where θ
is given by (16). Then, the Nash Demand Game possesses a unique logit
equilibrium. This equilibrium is symmetric across players.
Proof: Uniqueness follows directly from Theorem 2. To see why this unique
logit equilibrium must be symmetric, note the proof of Theorem 2 implies that
Te = Te1 = Te2 is a contraction. Hence Te must have a Þxed point F = TeF . But
this implies that (F, F ) = (Te(F ), Te(F )) = T (F, F ), and hence, by (12), (F, F )
is a logit equilibrium.
¥
Theorem 2 asserts that if players’ mistakes in computation are large in a
well-deÞned sense, then the logit equilibrium is unique. A natural question is
whether Theorem 2 does not establish uniqueness of logit equilibrium in the
NDG only for error parameters that are too large (relative to the payoff range
of the NDG) to be realistic. Computations of error parameters by Capra, Goeree and Holt give us some guidance. In the Traveler’s Dilemma game, for
instance, where players’ payoffs ranged from 80 to 200, these authors estimated
103
the common error parameter as 8.3 (see Capra et al., 1999). In a two-person
minimum effort coordination game with payoffs ranging from 110 to 170 this
estimate was 7.4 (see Goeree and Holt, 1999). These results suggest that in a
game with a given payoff range an error parameter can be considered “realistic”
if it is about ten times smaller than this range. For the NDG, then, a “realistic” µ is about 0.1. To relate this estimate to Theorem 2, suppose that both
bargainers are interested solely in payoffs: U1 (x) = U2 (x) = x. Then, Theorem
2 establishes uniqueness of logit equilibrium for µi > 3.84. Since 3.84 is far
greater than 0.1, Theorem 2 (and the ensuing corollary) is rather weak. This
weakness is a consequence of the upper bound on the right-hand side of (17).
Since this bound is quite coarse, we suppose that operator T is a contraction
on a greater range of error parameters than we were able to prove. We return
to the issue of uniqueness of logit equilibrium below.
Suppose now that U1 and U2 are differentiable. By Theorem 1, this assures
that logit-equilibrium cumulative density functions are twice differentiable. As
explained in section 2.3, this enables us to compute logit equilibrium by solving
a system of differential equations in a form of (8). Since
d e
0
(1 − x),
U (x) = Ui0 (x)F3−i (1 − x) − Ui (x)F3−i
dx i
this system is
£
¤
0 (1 − x)
µi Fi00 (x) = Fi0 (x) Ui0 (x)F3−i (1 − x) − Ui (x)F3−i
s.t. Fi (0) = 0, Fi (1) = 1
(19)
for i = 1, 2. Here the prime denotes the derivative with respect to x evaluated
on whatever argument appropriate.
System (19) is not a system of ordinary differential equations of the second
0
evaluated at 1−x rather than at
order because Fi00 (x) depends on F3−i and F3−i
x. Differential equations of this kind are called functional differential equations,
whose theory is less developed than the theory of their ordinary kin. Even less
developed is the theory of systems of functional differential equations.
Fortunately, one can convert (19) to a system of ordinary differential equations:
LEMMA 1: System (19) is equivalent to the system
£
¤
µ1 F 00 (x) = F 0 (x) U10 (x)(1 − G(x)) − U1 (x)G0 (x)
£
¤
µ2 G00 (x) = G0 (x) U2 (1 − x)F 0 (x) − U20 (1 − x)F (x)
(20)
subject to F (0) = G(0) = 0 and F (1) = G(1) = 1. The equivalence is
given by the functional transformation (F1 , F2 ) ↔ (F, G) with F (x) =
104
F1 (x) and G(x) = 1 − F2 (1 − x).11
The proof is a simple exercise in substituting and differentiating and is therefore
omitted.
Function G in Lemma 1 deÞnes a cumulative density function that describes
(distribution of) the complement to player 2’s demand, i.e., how much he offers
to player 1. Had we framed the NDG so that player 1 announces a demand
whereas player 2 announces an offer, we could have arrived at system (20)
directly.
System (20) does not have a closed-form solution even in the simple case
of U1 (x) = U2 (x) = x. By Theorem 1, though, we know that the solution to
this system exists for all pairs of error parameters µ1 , µ2 . Unfortunately, it is
not clear whether this solution is unique. The theory of systems of differential
equations is of no help here, since it deals mostly with initial-value problems
(see e.g., Bear, 1962), and (20) is a system of boundary-value problems. Even
the excellent book on boundary-value problems by Bailey et al. (1968) is silent
about systems of equations. We cannot therefore, at this point, prove uniqueness of logit equilibrium in the NDG for other error parameters than those
speciÞed by Theorem 2.12
We therefore tried to prove whether there is at least a unique symmetric
solution to (20) if U1 = U2 and µ1 = µ2 (by symmetric we mean a solution F ,
G such that F (x) = 1 − G(1 − x) for all x ∈ [0, 1]). Even this attempt was
futile, however. The “lens” technique (see, e.g., Anderson, Goeree and Holt,
forthcoming 2) which is commonly employed in this type of proof cannot be
used for (20) because this system is not symmetric.
Finally, following Anderson, Goeree and Holt (forthcoming 2) we tried to
prove that any density function describing logit equilibrium in the NDG is
11
Note that the substitution of F for F1 is only formal. We introduce it mainly to lighten
the notation in what follows.
12
The uniqueness of logit equilibrium for all pairs of error parameters would contrast sharply
with the continuum of Nash equilibria in this game. That a continuum of Nash equilibria needs
not be an obstacle to having a unique logit equilibrium was demonstrated by Anderson, Goeree
and Holt (2001) for minimum-effort coordination games. In the minimum-effort coordination
games, however, all Nash equilibria are symmetric, while in the NDG all but one Nash equilibria are asymmetric. Still, we believe that there is a unique logit equilibrium in the NDG, if not
for general utility functions and all pairs of error parameters, then at least for linear utilities
and a range of error parameters much larger than indicated by Theorem 2. Our conÞdence
here draws on a similarity between system (20) for linear utilities and a system describing logit
equilibrium in the imperfect price competition game (Capra et al., forthcoming), for which
uniqueness can be established. Numerical examples presented in the next section also support
our belief in uniqueness even in the case of small error parameters.
105
single-peaked. Once again, however, we were betrayed by the asymmetry of
(20).
4
The Examples
Since we were not able to characterize logit equilibrium in the NDG in general,
we now show examples of this equilibrium for various speciÞcations of players’
utilities and error parameters. First we show symmetric examples and then
asymmetric. All examples were calculated numerically by Maple V.
Figures 1, 2 and 3 illustrate a symmetric solution to system (20) in the case
when both players have the same utility U(x) = x and the same error parameter µ. The Þgures display both the cumulative density function F and the
corresponding density function f . The common error parameter µ diminishes
from Figure 1 to Figure 2 to Figure 3, taking values 1, 0.1, and 0.05.
Note that in all three cases the equilibrium density function f is, as we tried
to prove, single-peaked. The same will be true about other examples to follow.
Figure 1 shows that if µ = 1, the equilibrium density function f is almost
uniform (recall that the uniform distribution is the ‘limit’ logit equilibrium
density function for µ → +∞). This is not too surprising since µ = 1 is
quite large relative to the payoff range of the NDG (see the discussion following
Theorem 2). In any case this indicates that as µ → +∞ the convergence of the
logit-equilibrium densities to uniform densities is quite fast. This provides one
more reason why Theorem 2 ought to be considered weak.
Surprisingly, not even for µ = 0.05 were we able to Þnd multiple solutions to
(20). While this does not prove that for even lower error parameters multiple
logit equilibria cannot exist, it does support our conjecture that the range
of error parameters for which there exists a unique logit equilibrium is much
greater than shown in Theorem 2.
Figures 2 and 3 demonstrate how f accumulates as µ declines: in Figure 3
f(x) is almost zero for x ≤ 0.2 and x ≥ 0.6. This accumulation of probability
mass reßects the deÞnition of logit equilibrium: the smaller µ, the more sensitive
players are to differences in expected utilities from different demands, and so
the closer their strategy is to the best response. Interestingly, as µ declines, f
becomes markedly asymmetric. The asymmetry of f is especially conspicuous in
Figure 3, where f grows relatively slowly between 0.3 and 0.45 to drop sharply
to zero afterwards. This asymmetry reßects the asymmetric nature of Nash
equilibria in the NDG and is in accord with the conjecture we made in the
introduction that, when facing a boundedly rational opponent, players would
be cautious to make demands close to or larger than one half. The smaller the
106
µ, the less such caution is necessary and the more closely the f approximates
a demand of a half of the cake.
Table 1 provides a quantitative comparison of the logit equilibrium density
functions shown in Figures 1, 2 and 3.
Exp. value of
f
Peak of
f
Quantiles of
f
(10%, 50%, 90%)
Exp. equlib.
payoff
µ=1
0.499
0.481
0.112, 0.498, 0.887
0.170
µ = 0.1
0.431
0.456
0.264, 0.436, 0.573
0.316
µ = 0.05
0.451
0.486
0.383, 0.464, 0.499
0.436
Table 1: Characterization of symmetric logit equilibrium when both players
have linear utilities and the same error parameter µ.
The table adds details to our conjecture that boundedly rational players
would be cautious. If µ equals 1, both players make such large mistakes in
computation that they are almost indifferent to what to demand. The average
demand is 0.499, the median demand 0.498. Low rationality thus implies no
caution. If µ declines to 0.1, though, players are reluctant to make considerably
larger demands than one half: the average demand drops to 0.431, the median
to 0.436. Medium rationality thus implies some caution. If µ declines even more
to 0.05, however, players can be less cautious again since certainty about their
opponent’s demands rises. The demands move closer to one half from the left:
the average demand rises to 0.451, the median demand 0.464. High rationality
thus implies little need for caution.
Quantiles of f in column 4 of Table 1 show how the asymmetry of f grows
as µ decreases. They also demonstrate that the smaller the µ, the lower the
proportion of demands above one half. Column 5 of Table 1 shows that a
player’s expected payoff from bargaining grows as µ declines. This is natural,
since for small µ players’ demands are well coordinated and the cake is rarely
lost; rare also are situations when much of the cake remains unclaimed (when
µ is large, however, both these instances of miscoordination are frequent).
Figure 4 shows an asymmetric logit equilibrium in the case when both players have the same utility U(x) = x, but differ in their error parameters. Player
1 is twenty times ‘more rational’ than player 2: µ1 = 0.05, µ2 = 1. Again, only
one such equilibrium was found. Table 2 is an analog of Table 1 in this case.13
13
In this table, and others to follow, symbol f (x) is used somewhat loosely as a shorthand
for equilibrum “density function.” It refers to player 1’s equilibrium density function (which
is the hitherto used meaning of f (x)) as well as to player 2’s equibrium density function
(g(1 − x)). In contrast, Figure 4 shows function g(x). A peak of the density function of player
2 in Table 2, column 3, therefore does not correspond to the peak of g(x) in Figure 4.
107
Exp. value of
f
Peak of
f
Quantiles of
f
(10%, 50%, 90%)
Exp. equlib.
proÞt
player 1
0.490
0.489
0.305, 0.490, 0.676
0.240
player 2
0.485
0.396
0.110, 0.471, 0.884
0.153
Table 2: Characterization of asymmetric logit equilibrium when both players
have linear utilities but differ in their error parameters: µ1 = 0.05, µ2 = 1.
Here the equilibrium density function of player 1 is much ßatter than in
the case when he faces an equally rational opponent (see column 3 of Table 1).
Since player 2’s demands are very volatile, the probability of a miscoordination
is great and the differences in player 1’s expected payoffs from different demands
small. Player 1’s choice probabilities are therefore less dispersed–his density
function ßatter–than in case when his opponent’s error parameter is small.
Player 2’s equilibrium density function has a spread similar to that of the
density function in the Þrst row of Table 1, but it places more weight on smaller
demands. This might be interpreted so that it is worse for player 2 to be
confronted with a more rational opponent, than with the selfsame opponent.
The last column of Table 2 supports this interpretation: player 2’s expected
payoff against a more rational player 1 (0.153) is lower than his payoff against
an equally irrational player 1 (0.170; see the Þrst row of Table 1). It cannot be
said, however, player 2 is exploited by more rational player 1. Player 1 himself is
better off when facing an equally rational opponent (compare expected payoffs
in the Þrst row of Table 2 and the last row of Table 1). In fact, as far as loss in
expected payoff is concerned, player 1 is hurt more by player 2’s irrationality
than player 2 by player 1’s rationality.
Table 1 and Figures 1, 2 and 3 demonstrate that as µ declines, the logit
equilibrium (which, as said, appears to be unique), approaches a fair division
of the cake ( 12 , 12 ). If both players have identical utility functions, this division
coincides with both the Nash bargaining solution (Nash, 1950) and the RaiffaKalai-Smorodinsky bargaining solution (henceforth RKS solution; see Raiffa,
1953, and Kalai and Smorodinsky, 1975). It is therefore not clear whether for
small µ the logit equilibrium approximates the former or the latter bargaining
solution.
To discriminate between these two hypotheses, we calculated logit equilibrium in two cases when players’ utility functions differ. In both cases µ1
and µ2 were set as identical and equal to 0.05, the smallest error parameter
Maple V could handle. In the Þrst case players’ utilities were chosen as follows:
U1 (x) = log(1 + 21.933x) and U2 (x) = x. The rather weird choice of U1 guaran-
108
tees a nice Nash bargaining solution (0.3, 0.7).14 The RKS bargaining solution
is (0.226, 0.773).
As before, we found only one logit equilibrium. Figure 5 and Table 3 describe
the density functions of this equilibrium in detail.
Exp. value of
f
Peak of
f
Quantiles of
f
(5%, 50%, 95%)
Exp. equlib.
proÞt
player 1
0.293
0.306
0.262, 0.289, 0.296
0.291
player 2
0.641
0.685
0.580, 0.667, 0.694
0.641
Table 3: Characterization of asymmetric logit equilibrium when both players
have the same error parameters µ1 = 0.05 but differ in their utility functions:
U1 (x) = log(1 + 21.933x), U2 (x) = x.
Both the Þgure and the table provide strong support for the hypothesis
that for small µ logit equilibrium approximates the Nash bargaining solution.
This is best seen at players’ expected or median payoffs, which are much closer
to their Nash-bargaining-solution shares than to their RKS shares. The RKS
bargaining solution therefore cannot be considered a point approximated by the
logit equilibrium.
The second example points to the same conclusion even more sharply. In
√
this example, players’ utilities were chosen as U1 (x) = 1 + 20x − 1, U2 (x) =
x. This choice guarantees the Nash bargaining solution (0.4, 0.6); the RKS
solution is (0.126, 0.874). Table 4 and Figure 6 show the logit equilibrium in
this case. Again, neither player made demands close to his RKS share with
high probability, but both players demanded on average almost their Nashbargaining-solution shares.
Exp. value of
f
Peak of
f
Quantiles of
f
(10%, 50%, 90%)
Exp. equlib.
proÞt
player 1
0.389
0.400
0.370, 0.393, 0.403
0.387
player 2
0.546
0.591
0.481, 0.562, 0.591
0.544
Table 4: Characterization of asymmetric logit equilibrium when both players
have the same error parameters µ1 = 0.05 but differ in their utility functions:
√
U1 (x) = 1 + 20x − 1, U2 (x) = x.
Both examples are quite persuasive. The Nash bargaining solution seems to
be an accumulation point of logit equilibrium when players’ error parameters
14
U1 (x) was determined ex post after the Nash bargaining solution was Þxed at (0.3, 0.7)
and U2 (x) put equal to x.
109
tend to zero. Since there may be more than one logit equilibrium for small error
parameters, however, there may be more than one such accumulation point. In
spite of this difficulty, a rigorous hypothesis about the limit behavior of logit
equilibrium can be framed with the aid of logit equilibrium correspondence,
as done by McKelvey and Palfrey (1995) for Þnite-strategy games. The logit
equilibrium correspondence Λ : R+ → 2M×M is given as
Λ(µ) ≡ {(F1 , F2 ) ∈ M × M | (F1 , F2 ) = Tµ (F1 , F2 )} ,
where Tµ stands for operator T deÞned by (10), when µ1 = µ2 = µ. Then,
the hypothesis is that there is a unique branch in the graph of Λ that starts
at a pair of uniform cumulative density functions at µ = +∞ (the so-called
centroid) and converges to the Nash bargaining solution as µ tends to zero.
We see a theoretical investigation of this hypothesis as a salient task for future
research.
5
Conclusion
In this note we presented a boundedly rational analysis of the Nash Demand
Game based on the concept of quantal response equilibrium (logit equilibrium).
We reviewed the concept and discussed its evolutive support. We then showed
that logit equilibrium exists in the Nash Demand Game and it is unique if
neither bargainer is too close to the ideal of a “perfectly rational man.” If,
moreover, both bargainers have the same utility function, then the logit equilibrium in the Nash Demand Game is symmetric.
We also computed numerical examples of logit equilibria in the Nash Demand Game for various speciÞcations of players’ utilities and various degrees
of their ‘irrationality’ (error parameters). These examples reßected our conjecture that boundedly rational players would be cautious in their demands,
since overreaching oneself in bargaining may be more costly than leaving a part
of the ‘cake’ unclaimed. More importantly, the examples supported the hypothesis that, as players’ error parameters tend to zero at the same rate, logit
equilibrium in the Nash Demand Game selects a unique bargaining solution,
namely the Nash bargaining solution. This Þnding corroborates the robustness
of the Nash bargaining solution to “trembles,” as evidenced by Young (1993)
and Binmore and Dasgupta (1987).
6
References
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110
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112
F igu res
1.2
f(x)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
x
1.2
F(x)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
x
F igure 1: Log it equilibrium density function (to p) and cumul ative density functio n (bo ttom) when b oth pl ayers have lin ear
u tilities and ¹1 = ¹2 = 1 .
1 13
4.5
f(x)
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
x
1.2
F(x)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
x
F igure 2: Log it equilibrium density function (to p) and cumul ative density functio n (bo ttom) when b oth pl ayers have lin ear
u tilities and ¹1 = ¹2 = 0:1.
1 14
14
f(x)
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
x
1.2
F(x)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
x
F igure 3: Log it equilibrium density function (to p) and cumul ative density functio n (bo ttom) when b oth pl ayers have lin ear
u tilities and ¹1 = ¹2 = 0:05.
1 15
3
f(x), g(x)
f
2.5
g
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
1.2
F(x), G(x)
1
0.8
0.6
F
G
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
F igure 4 : Log it equ ilibriu m d ensity functio ns (top) a nd cumul ative density functions (bottom) when b oth players have lin ear
u tilities and ¹1 = 0:05 , ¹2 = 1.
1 16
1
50
f(x), g(x)
45
40
f(x)
35
g(x)
30
25
20
15
10
5
0
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x
1.2
F(x), G(x)
1
F
0.8
G
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
F igure 5: Logit equilibrium density functions (top) a nd cumulative density fu nctions (b ottom) w hen U1 (x) = log(1 + 21:933x),
U2 (x) = x, and ¹1 = ¹2 = 0:05.
1 17
1
60
f(x), g(x)
50
f(x)
40
g(x)
30
20
10
0
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x
1.2
F(x), G(x)
1
F
0.8
G
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
F igure 6 : Log it equ ilibriu m d ensity functio ns (top) a nd cumul ative den sity functions (b ottom ) when U1(x) =
U2 (x) = x, and ¹1 = ¹2 = 0:05.
1 18
p
1 + 20x ¡ 1,
1